Finite Group Databasebeta

← C31×#9999 #21466758

Subgroups — C31×#9999 #21466758

Order 7936.

554 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C31×#3103C211yesmaximal, characteristic
2C31×#3037C211yesmaximal, characteristic
3C31×#2581C211yesmaximal, characteristic, complemented
4C31×#1517C211yesmaximal, characteristic
5C31×#1487C211yesmaximal, characteristic, complemented
6C31×#1479C211yesmaximal, characteristic, complemented
7C31×#1286C211yesmaximal, characteristic, complemented
8C2×C31×#137—21no—
9C31×#532C2211yescharacteristic
10C2×C4×Q8×C31C2211yescharacteristic
11C2×C4×Q8×C31C2211yescharacteristic
12C31×#481—21no—
13C31×#475—21no—
14C31×#464—22no—
15C31×#462—21no—
16C31×#460—21no—
17C31×#460—21no—
18C31×#448—21no—
19C4×SD16×C31—22no—
20C31×#431C2211yescharacteristic
21C31×#431C2211yescharacteristic
22C31×#423C2211yescharacteristic
23C31×#417—21no—
24C31×#400—21no—
25C31×#395—21no—
26C31×#391—21no—
27C31×#389—21no—
28C31×#387—22no—
29C31×#382C2211yescharacteristic
30C31×#327—22no—
31C31×Q8⋊C8—22no—
32C31×Q8⋊C8—22no—
33C2×#41×C31—21no—
34C22×Q8×C31D411yescharacteristic
35C22×Q8×C31D411yescharacteristic
36C31×#137—42no—
37C31×#137—42no—
38C2×Q16×C31—22no—
39C2×SD16×C31—22no—
40C2×SD16×C31—22no—
41C2×M16×C31—21no—
42C31×C4⋊Q8D411yescharacteristic
43C31×#127D411yescharacteristic
44C31×#124—21no—
45C31×#121—21no—
46C31×#121—21no—
47C4×Q8×C31—21no—
48C4×Q8×C31—21no—
49C4×Q8×C31—21no—
50C4×Q8×C31—21no—
51C4×Q8×C31—21no—
52C4×Q8×C31—21no—
53C4×Q8×C31—21no—
54C4×Q8×C31—21no—
55C4×D4×C31—21no—
56C4×D4×C31—21no—
57C2×C4⋊C4×C31—22no—
58C2×C4⋊C4×C31—21no—
59C2×C4⋊C4×C31—21no—
60C2×C4⋊C4×C31—21no—
61C2×C4⋊C4×C31D411yescharacteristic
62C2×C4⋊C4×C31D411yescharacteristic
63C2×C42×C31—21no—
64C2×C42×C31—21no—
65C2×C42×C31C2311yescharacteristic
66C31×#106—22no—
67C31×C4⋊C8—21no—
68C31×C4⋊C8—21no—
69C31×C4⋊C8—21no—
70C31×C4⋊C8D412yes—
71C31×#103—22no—
72C31×#103—22no—
73C31×#102—22no—
74C31×#102—22no—
75C31×C22⋊C8—21no—
76C31×C22⋊C8—21no—
77C4×C8×C31—21no—
78C4×C8×C31D412yes—
79C31×#95—22no—
80C31×#95—22no—
81C31×#95—22no—
82C31×#95—21no—
83C31×#95D412yes—
84#41×C31—41no—
85#41×C31—41no—
86#41×C31—41no—
87#41×C31—41no—
88C2×Q8×C31—21no—
89C2×Q8×C31—21no—
90C2×Q8×C31—21no—
91C2×Q8×C31—21no—
92C2×Q8×C31—21no—
93C2×Q8×C31—21no—
94C2×Q8×C31—21no—
95C2×Q8×C31—21no—
96C2×Q8×C31—42no—
97C2×Q8×C31SD1612yescomplemented
98C2×D4×C31—21no—
99C2×D4×C31—21no—
100C2×D4×C31—21no—
101C2×D4×C31—41no—
102C22×C4×C31—21no—
103C22×C4×C31—21no—
104C22×C4×C31—22no—
105C22×C4×C31—21no—
106C22×C4×C31—21no—
107C22×C4×C31—41no—
108C22×C4×C31C2×D411yescharacteristic
109C22×C4×C31C2×D411yescharacteristic
110C22×C4×C31C2×D411yescharacteristic
111Q16×C31—44no—
112SD16×C31—44no—
113SD16×C31—44no—
114M16×C31—41no—
115M16×C31—41no—
116C2×C8×C31—22no—
117C2×C8×C31—22no—
118C2×C8×C31—41no—
119C4⋊C4×C31—21no—
120C4⋊C4×C31—21no—
121C4⋊C4×C31—21no—
122C4⋊C4×C31—21no—
123C4⋊C4×C31—21no—
124C4⋊C4×C31—21no—
125C4⋊C4×C31—44no—
126C4⋊C4×C31—41no—
127C4⋊C4×C31—41no—
128C4⋊C4×C31—41no—
129C4⋊C4×C31—41no—
130C4⋊C4×C31—41no—
131C4⋊C4×C31—41no—
132C22⋊C4×C31—41no—
133C22⋊C4×C31—41no—
134C42×C31—41no—
135C42×C31—41no—
136C42×C31—41no—
137C42×C31—41no—
138C42×C31C2×D411yescharacteristic
139C42×C31C2×D411yescharacteristic
140C42×C31C2×D411yescharacteristic
141C42×C31#4111yescharacteristic
142#9999C3111yesmaximal, characteristic, direct product factor, Sylow
143C23×C31—41no—
144C23×C31#12011yescharacteristic
145Q8×C31—22no—
146Q8×C31—22no—
147Q8×C31—22no—
148Q8×C31—22no—
149Q8×C31—41no—
150Q8×C31—41no—
151Q8×C31—41no—
152Q8×C31—41no—
153Q8×C31—41no—
154Q8×C31—41no—
155D4×C31—42no—
156D4×C31—41no—
157D4×C31—41no—
158D4×C31—81no—
159D4×C31—81no—
160D4×C31—81no—
161C2×C4×C31—22no—
162C2×C4×C31—22no—
163C2×C4×C31—41no—
164C2×C4×C31—41no—
165C2×C4×C31—41no—
166C2×C4×C31—41no—
167C2×C4×C31—41no—
168C2×C4×C31—41no—
169C2×C4×C31—42no—
170C2×C4×C31—42no—
171C2×C4×C31—42no—
172C2×C4×C31—42no—
173C2×C4×C31—41no—
174C2×C4×C31—41no—
175C2×C4×C31—41no—
176C2×C4×C31—42no—
177C2×C4×C31—81no—
178C2×C4×C31—81no—
179C2×C4×C31#12011yescharacteristic
180C2×C4×C31#12011yescharacteristic
181C2×C4×C31#12111yescharacteristic
182C2×C4×C31#12111yescharacteristic
183C2×C4×C31#12111yescharacteristic
184C2×C4×C31#12711yescharacteristic
185C2×C4×C31C2×SD1611yescharacteristic
186C2×C4×C31#13511yescharacteristic
187C2×C4×C31AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C31#13711yescharacteristic
189C8×C31—42no—
190C8×C31—42no—
191C8×C31—81no—
192#3103C2×C3111yescharacteristic
193#3037C2×C3111yescharacteristic
194#2581C2×C3111yescharacteristic, complemented
195#1517C2×C3111yescharacteristic
196#1487C2×C3111yescharacteristic, complemented
197#1479C2×C3111yescharacteristic, complemented
198#1286C2×C3111yescharacteristic, complemented
199C22×C31—21no—
200C22×C31—21no—
201C22×C31—81no—
202C22×C31—81no—
203C22×C31—81no—
204C22×C31#39211yescharacteristic
205C22×C31#44811yescharacteristic
206C22×C31#44911yescharacteristic
207C4×C31—21no—
208C4×C31—21no—
209C4×C31—21no—
210C4×C31—21no—
211C4×C31—42no—
212C4×C31—42no—
213C4×C31—82no—
214C4×C31—81no—
215C4×C31—81no—
216C4×C31—81no—
217C4×C31#45011yescharacteristic
218C4×C31#45211yescharacteristic
219C4×C31#46511yescharacteristic
220C4×C31#46811yescharacteristic
221C2×#137—21no—
222#532C22×C3111yescharacteristic
223C2×C4×Q8C22×C3111yescharacteristic
224C2×C4×Q8C22×C3111yescharacteristic
225#481—21no—
226#475—21no—
227#464—22no—
228#462—21no—
229#460—21no—
230#460—21no—
231#448—21no—
232C4×SD16—22no—
233#431C22×C3111yescharacteristic
234#431C22×C3111yescharacteristic
235#423C22×C3111yescharacteristic
236#417—21no—
237#400—21no—
238#395—21no—
239#391—21no—
240#389—21no—
241#387—22no—
242#382C22×C3111yescharacteristic
243#327—22no—
244Q8⋊C8—22no—
245Q8⋊C8—22no—
246C2×C31—21no—
247C2×C31—21no—
248C2×C31—161no—
249C2×C31#200011yescharacteristic
250C2×C31#200911yescharacteristic
251C2×C31#201011yescharacteristic
252C2×#41—21no—
253C22×Q8D4×C3111yescharacteristic
254C22×Q8D4×C3111yescharacteristic
255#137—42no—
256#137—42no—
257C2×Q16—22no—
258C2×SD16—22no—
259C2×SD16—22no—
260C2×M16—21no—
261C4⋊Q8D4×C3111yescharacteristic
262#127D4×C3111yescharacteristic
263#124—21no—
264#121—21no—
265#121—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×D4—21no—
275C4×D4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4D4×C3111yescharacteristic
281C2×C4⋊C4D4×C3111yescharacteristic
282C2×C42—21no—
283C2×C42—21no—
284C2×C42C23×C3111yescharacteristic
285#106—22no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8D4×C3112yes—
290#103—22no—
291#103—22no—
292#102—22no—
293#102—22no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C4×C8—21no—
297C4×C8D4×C3112yes—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—21no—
302#95D4×C3112yes—
303C31#999911yescharacteristic, direct product factor, Sylow
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—42no—
317C2×Q8SD16×C3112yescomplemented
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—41no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4C2×D4×C3111yescharacteristic
329C22×C4C2×D4×C3111yescharacteristic
330C22×C4C2×D4×C3111yescharacteristic
331Q16—44no—
332SD16—44no—
333SD16—44no—
334M16—41no—
335M16—41no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—41no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—44no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42C2×D4×C3111yescharacteristic
359C42C2×D4×C3111yescharacteristic
360C42C2×D4×C3111yescharacteristic
361C42#41×C3111yescharacteristic
362C23—41no—
363C23C31×#12011yescharacteristic
364Q8—22no—
365Q8—22no—
366Q8—22no—
367Q8—22no—
368Q8—41no—
369Q8—41no—
370Q8—41no—
371Q8—41no—
372Q8—41no—
373Q8—41no—
374D4—42no—
375D4—41no—
376D4—41no—
377D4—81no—
378D4—81no—
379D4—81no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4C31×#12011yescharacteristic
399C2×C4C31×#12011yescharacteristic
400C2×C4C31×#12111yescharacteristic
401C2×C4C31×#12111yescharacteristic
402C2×C4C31×#12111yescharacteristic
403C2×C4C31×#12711yescharacteristic
404C2×C4C2×SD16×C3111yescharacteristic
405C2×C4C31×#13511yescharacteristic
406C2×C4C31×AGL(1,ℤ/8)11yescharacteristic
407C2×C4C31×#13711yescharacteristic
408C8—42no—
409C8—42no—
410C8—81no—
411C22—21no—
412C22—21no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22C31×#39211yescharacteristic
417C22C31×#44811yescharacteristic
418C22C31×#44911yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—42no—
424C4—42no—
425C4—82no—
426C4—81no—
427C4—81no—
428C4—81no—
429C4C31×#45011yescharacteristic
430C4C31×#45211yescharacteristic
431C4C31×#46511yescharacteristic
432C4C31×#46811yescharacteristic
433C2—21no—
434C2—21no—
435C2—161no—
436C2C31×#200011yescharacteristic
437C2C31×#200911yescharacteristic
438C2C31×#201011yescharacteristic
subgroupG/H
center, socleC22×C31#392
commutator, Frattini, cosocleC2×C42C23×C31
prime radicalC31×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C31×#2581C2semidirect
5. C31×#1487C2semidirect
6. C31×#1479C2semidirect
7. C31×#1286C2semidirect
97. C2×Q8×C31SD16semidirect
142. #9999C31direct (N × H)
194. #2581C2×C31semidirect
196. #1487C2×C31semidirect
197. #1479C2×C31semidirect
198. #1286C2×C31semidirect
303. C31#9999direct (N × H)
317. C2×Q8SD16×C31semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C31×#31032
C31×#30372
C31×#25812
C31×#15172
C31×#14872
C31×#14792
C31×#12862
#999931